Higher differentiability for solutions to a class of obstacle problems

Abstract

We establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the formwhere A is a p-harmonic type operator, ψ ∈ W 1,p (Ω) is a fixed function called obstacle ande. in Ω} is the class of the admissible functions. We prove that an extra differentiability assumption on the gradient of the obstacle transfers to Du with no losses in the natural exponent of integrability, provided the partial map x → A(x, ξ) possesses a suitable differentiability property measured or in the scale of the Sobolev space W 1,n or in that of the critical Besov-Lipschitz spaces B α n α ,q , for a suitable 1 ≤ q ≤ +∞.2000 Mathematics Subject Classification. 35J87, 49J40; 47J20. Key words and phrases. Variational inequalities, obstacle problems, higher differentiability. The work of Michela Eleuteri is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), through the projects GNAMPA 2016 "Regolarità e comportamento asintotico di soluzioni di equazioni paraboliche" (coord. Prof. S. Polidoro) and GNAMPA 2017 "Regolarità per problemi variazionali d'ostacolo e liberi" (coord. Prof. M. Focardi). The work of Antonia Passarelli di Napoli is supported by GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica), through the projects GNAMPA 2016 "Problemi di Regolaritá nel Calcolo delle Variazioni e di Approssimazione" (coord. Prof. M. Carozza) and GNAMPA 2017 "Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all'imaging" (coord. Dott. D. Costarelli). The work of the authors is also supported by the University of Modena and Reggio Emilia through the project FAR2015 "Equazioni differenziali: problemi evolutivi, variazionali ed applicazioni" (coord. Prof. S. Polidoro). This reaserch started while A. Passarelli di Napoli was visiting the University of Modena and Reggio Emilia . The hospitality of this Institution is warmly aknowledged.[42] G. Stampacchia: Formes bilineaires coercivitives sur les ensembles convexes,

Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence

“…In this context see [6], [17], [10] and recently [13], [14] and [9]. The Sobolev dependence on x recently has been considered in [22], [1] and for obstacle problems in [16].…”

“…Finally we need to estimate the sixth integral in (16). Let us observe that we want to consider growth conditions only at infinity, therefore we need to overcome the difficulty due to the presence of the term Φ ′ in this sixth integral.…”

Regularity results for a class of obstacle problems with p, q−growth conditions

“…We'd like to mention that Theorem 1.1 has been already employed in [8] to establish an higher differentiability result under weaker assumptions on the integrand F and on the obstacle ψ with respect to those in [23].…”

“…In particular we remark that in [27] a lot of effort has been employed to identify the Radon measure and the authors explicitly say that this procedure could be significantly simplified if we would have a priori proved higher differentiability for local minimizers of the obstacle problem. But in a recent paper ( [23]), the authors were able to establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems (involving p−harmonic operators) assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. The use of this result simplifies the procedure outlined in [27] and has been employed in [1] to obtain the local Lipschitz continuity of solutions to (1.3) under standard growth conditions and with Lipschitz continuous coefficients.…”

“…This phenomenon has been widely investigated in case of non constrained problems (see for example [29,30,[39][40][41]) and more recently also in case of obstacle problems ( [23,28]).…”